A326080 - OEIS

A326080

Number of subsets of {1..n} containing the sum of every subset whose sum is <= n.

1, 2, 4, 7, 12, 19, 31, 47, 73, 110, 168, 247, 375, 546, 817, 1193, 1769, 2552, 3791, 5445, 8012, 11517, 16899, 24144, 35391, 50427, 73614, 104984, 152656, 216802, 315689, 447473, 648813, 920163, 1332991, 1884735, 2728020, 3853437, 5568644, 7868096, 11347437

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OFFSET

0,2

COMMENTS

Equivalently, a(n) is the number of subsets of {1..n} containing the sum of any two distinct elements whose sum is <= n.

The summands must be distinct. The case where they are allowed to be equal is A326083.

If A151897 counts sum-free sets, this sequence counts sum-closed sets. This is different from sum-full sets (A093971).

LINKS

Table of n, a(n) for n=0..40.

EXAMPLE

The a(0) = 1 through a(5) = 19 subsets:

{} {} {} {} {} {}

{1} {1} {1} {1} {1}

{2} {2} {2} {2}

{1,2} {3} {3} {3}

{1,3} {4} {4}

{2,3} {1,4} {5}

{1,2,3} {2,3} {1,5}

{2,4} {2,4}

{3,4} {2,5}

{1,3,4} {3,4}

{2,3,4} {3,5}

{1,2,3,4} {4,5}

{1,4,5}

{2,3,5}

{2,4,5}

{3,4,5}

{1,3,4,5}

{2,3,4,5}

{1,2,3,4,5}

The a(6) = 31 subsets:

{} {1} {1,6} {1,5,6} {1,4,5,6} {1,3,4,5,6} {1,2,3,4,5,6}

{2} {2,5} {2,3,5} {2,3,5,6} {2,3,4,5,6}

{3} {2,6} {2,4,6} {2,4,5,6}

{4} {3,4} {2,5,6} {3,4,5,6}

{5} {3,5} {3,4,5}

{6} {3,6} {3,4,6}

{4,5} {3,5,6}

{4,6} {4,5,6}

{5,6}

MATHEMATICA

Table[Length[Select[Subsets[Range[n]], SubsetQ[#, Select[Plus@@@Subsets[#, {2}], #<=n&]]&]], {n, 0, 10}]

PROG

(PARI)

a(n)={

my(recurse(k, b)=

if( k > n, 1,

my(t=self()(k + 1, b + (1<<k)));

for(i=1, (k-1)\2, if(bittest(b, i) && bittest(b, k-i), return(t)));

t + self()(k + 1, b) )

);

recurse(1, 0);

} \\ Andrew Howroyd, Aug 30 2019

CROSSREFS

Cf. A007865, A050291, A051026, A054519, A085489, A093971, A103580, A120641, A151897, A326020, A326023, A326076, A326083.

Sequence in context: A333148 A343661 A342229 * A287525 A244472 A279890

Adjacent sequences: A326077 A326078 A326079 * A326081 A326082 A326083

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 05 2019

EXTENSIONS

Terms a(21) and beyond from Andrew Howroyd, Aug 30 2019

STATUS

approved